11. If
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
31.
32.
33.
35.
37.
39.
41.
42.
43. (b) The identically zero function
44. (a) The graphs (figure below) of typical solutions with
(b) The graphs (figure below) of typical solutions with
45.
46.
47. (a)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
19.
20.
21.
22.
23.
24.
25. The car stops when
26. (a)
27.
28.
29. After 10 seconds the car has traveled 200 ft and is traveling at 70 ft/s.
30.
31.
32. 60 m
33.
34. 460.8 ft
36. About 13.6 ft
37. 25 (mi)
38. 1:10 pm
39. 6 mph
40. 2.4 mi
41.
42. 25 mi
43. Time:
Distance:
44. About 54 mi/h
11. A unique solution exists in some neighborhood of
12. A unique solution exists in some neighborhood of
13. A unique solution exists in some neighborhood of
14. Existence but not uniqueness is guaranteed in some neighborhood of
15. Neither existence nor uniqueness is guaranteed in any neighborhood of
16. A unique solution exists in some neighborhood of
17. A unique solution exists in some neighborhood of
18. Neither existence nor uniqueness is guaranteed.
19. A unique solution exists in some neighborhood of
20. A unique solution exists in some neighborhood of
21. Your figure should suggest that
22.
23. Your figure should suggest that
24.
25. Your figure should suggest that the limiting velocity is about 20 ft/sec (quite survivable) and that the time required to reach 19 ft/sec is a little less than 2 seconds. An exact solution gives
26. A figure suggests that there are 40 deer after about 60 months; a more accurate value is
27. The initial value problem
28. The initial value problem
29. The initial value problem
30. The initial value problem
31. The initial value problem
32. The initial value problem
33. The initial value problem
34. (a) If
If
35. (a) If
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29. (a) General solution
30. General solution
31. Separation of variables gives the same general solution
32. General solution
(a) No solution if
33. About 51840 persons
34.
35. About 14735 years
36. Age about 686 years
37. $21103:48
38. $44.52
39. 2585 mg
40. About 35 years
41. About
42. About 1.25 billion years
43. After a total of about 63 min have elapsed
44. About 2.41 minutes
45. (a) 0.495 m; (b
46. (a) About 9.60 inches; (b) About 18,200 ft
47. After about 46 days
48. About 6 billion years
49. After about 66 min 40 s
50. (a)
51. (a)
52. About 120 thousand years ago
53. About 74 thousand years ago
54. 3 hours
55. 972 s
56. At time
58. 1:20 p.m.
59. (a)
60. About 6 min 3 sec
61. Approximately 14 min 29 s
62. The tank is empty about 14 seconds after 2:00 p.m.
63. (a) 1:53:34 p.m.; (b)
64.
65. At approximately 10:29 a.m.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
30.
29.
32. (a)
33. After about 7 min 41 s
34. About 22.2 days
35. About 5.5452 years
36. (a)
37. 393:75 lb
38. (a)
39. (b)
41. (b) Approximately $1,308,283
43.
44. 3.99982, 4.00005, 4.00027, 4.00050, 4.00073
45.
46.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
58.
59.
60.
61.
64.
65.
69. Approximately 3.68 mi
1. Linear:
2. Separable:
3. Homogeneous:
4. Exact:
5. Separable:
6. Separable:
7. Linear:
8. Homogeneous:
9. Bernoulli:
10. Separable:
11. Homogeneous:
12. Exact:
13. Separable:
14. Homogeneous:
15. Linear:
16. Substitution:
17. Exact:
18. Homogeneous:
19. Separable:
20. Linear:
21. Linear:
22. Bernoulli:
23. Exact:
24. Separable:
25. Linear:
26. Exact:
27. Bernoulli:
28. Linear:
29. Linear:
30. Substitution:
31. Separable and linear
32. Separable and Bernoulli
33. Exact and homogeneous
34. Exact and homogeneous
35. Separable and linear
36. Separable and Bernoulli
1.
2.
3.
4.
5.
6.
7.
8.
9. 484
10. 20 weeks
11. (b)
12.
13.
14.
16. About 27.69 months
17. About 44.22 months
19. About 24.41 months
20. About 42.12 months
21.
22. About 34.66 days
23. (a)
24. About 9.24 days
25. (a)
26.
27. (a)
28. (a) The alligators eventually die out.(b) Doomsday occurs after about 9 years 2 months.
29. (a)
31.
37.
38.
39.
1. Unstable critical point:
2. Stable critical point:
3. Stable critical point:
4. Stable critical point:
5. Stable critical point:
6. Stable critical point:
7. Semi-stable (see Problem 18 ) critical point:
8. Semi-stable critical point:
9. Stable critical point:
10. Stable critical point:
11. Unstable critical point:
12. Stable critical point:
For each of Problems 13 through 18 we show a plot of slope field and typical solution curves. The equilibrium solutions of the given differential equation are labeled, and the stability or instability of each should be clear from the picture.
19. There are two critical points if
20. There are two critical points if
1. Approximately 31.5 s
3.
5.
7. (a) 100 ft/sec; (b) about 23 sec and 1403 ft to reach 90 ft/sec
8. (a) 100 ft/sec; (b) about 14.7 sec and 830 ft to reach 90 ft/sec
9. 50 ft/s
10. About 5 min 47 s
11. Time of fall: about 12.5 s
12. Approximately 648 ft
19. Approximately 30.46 ft/s; exactly 40 ft/s
20. Approximately 277.26 ft
22. Approximately 20.67 ft/s; about 484.57 s
23. Approximately 259.304 s
24. (a) About 0:88 cm; (b) about 2:91 km
25. (b) About 1.389 km/sec; (c)
26. Yes
28. (b) After about
29. About 51.427 km
30. Approximately 11.11 km/sec (as compared with the earth’s escape velocity of about 11.18 km/sec).
In Problems 1 through 10 we round off the indicated values to 3 decimal places.
1. Approximate values 1.125 and 1.181; true value 1.213
2. Approximate values 1.125 and 1.244; true value 1.359
3. Approximate values 2.125 and 2.221; true value 2.297
4. Approximate values 0.625 and 0.681; true value 0.713
5. Approximate values 0.938 and 0.889; true value 0.851
6. Approximate values 1.750 and 1.627; true value 1.558
7. Approximate values 2.859 and 2.737; true value 2.647
8. Approximate values 0.445 and 0.420; true value 0.405
9. Approximate values 1.267 and 1.278; true value 1.287
10. Approximate values 1.125 and 1.231; true value 1.333
Problems 11 through 24 call for tables of values that would occupy too much space for inclusion here. In Problems 11 through 16 we give first the final x-value, next the corresponding approximate x-values obtained with step sizes
11.
12. 1.0, 2.9864, 2.9931, 3.0000
13. 2.0, 4.8890, 4.8940, 4.8990
14. 2.0, 3.2031, 3.2304, 3.2589
15. 3.0, 3.4422, 3.4433, 3.4444
16. 3.0, 8.8440, 8.8445, 8.8451
In Problems 17 through 24 we give first the final x-value and then the corresponding approximate y-values obtained with step sizes
17. 1.0, 0.2925, 0.3379, 0.3477, 0.3497
18. 2.0, 1.6680, 1.6771, 1.6790, 1.6794
19. 2.0, 6.1831, 6.3653, 6.4022, 6.4096
20. 2.0,
21. 2.0, 2.8508, 2.8681, 2.8716, 2.8723
22. 2.0, 6.9879, 7.2601, 7.3154, 7.3264
23. 1.0, 1.2262, 1.2300, 1.2306, 1.2307
24. 1.0, 0.9585, 0.9918, 0.9984, 0.9997
25. With both step sizes
26. With both step sizes
27. With successive step sizes
28. With successive step sizes
| x | y |
y |
y |
|---|---|---|---|
| 1.0000 | 1.0000 | 1.0000 | |
| 1.0472 | 1.0512 | 1.0521 | |
| 1.1213 | 1.1358 | 1.1390 | |
| 1.2826 | 1.3612 | 1.3835 | |
| 0.2 | 0.8900 | 1.4711 | 0.8210 |
| 0.5 | 0.7460 | 1.2808 | 0.7192 |
| x | y |
y |
|---|---|---|
| 1.8 | 2.8200 | 4.3308 |
| 1.9 | 3.9393 | 7.9425 |
| 2.0 | 5.8521 | 28.3926 |
| x | y |
y |
|---|---|---|
| 0.7 | 4.3460 | 6.4643 |
| 0.8 | 5.8670 | 11.8425 |
| 0.9 | 8.3349 | 39.5010 |
| x | Improved Euler y | Actual y |
|---|---|---|
| 0.1 | 1.8100 | 1.8097 |
| 0.2 | 1.6381 | 1.6375 |
| 0.3 | 1.4824 | 1.4816 |
| 0.4 | 1.3416 | 1.3406 |
| 0.5 | 1.2142 | 1.2131 |
Note: In Problems 2 through 10, we give the value of x, the corresponding improved Euler value of y, and the true value of y.
2. 0.5, 1.3514, 1.3191
3. 0.5, 2.2949, 2.2974
4. 0.5, 0.7142, 0.7131
5. 0.5, 0.8526, 0.8513
6. 0.5, 1.5575, 1.5576
7. 0.5, 2.6405, 2.6475
8. 0.5, 0.4053, 0.4055
9. 0.5, 1.2873, 1.2874
10. 0.5, 1.3309, 1.3333
In Problems 11 through 16 we give the final value of x, the corresponding values of y with
11. 1.0,
12. 1.0, 2.99995, 2.99999, 3.00000
13. 2.0, 4.89901, 4.89899, 4.89898
14. 2.0, 3.25847, 3.25878, 3.25889
15. 3.0, 3.44445, 3.44445, 3.44444
16. 3.0, 8.84511, 8.84509, 8.84509
In Problems 17 through 24 we give the final value of x and the corresponding values of y for
17. 1.0, 0.35183, 0.35030, 0.35023, 0.35023
18. 2.0, 1.68043, 1.67949, 1.67946, 1.67946
19. 2.0, 6.40834, 6.41134, 6.41147, 6.41147
20. 2.0,
21. 2.0, 2.87204, 2.87245, 2.87247, 2.87247
22. 2.0, 7.31578, 7.32841, 7.32916, 7.32920
23. 1.0, 1.22967, 1.23069, 1.23073, 1.23073
24. 1.0, 1.00006, 1.00000, 1.00000, 1.00000
25. With both step sizes
26. With both step sizes
27. With successive step sizes
28. With successive step sizes
29. Impact speed approximately 43.22 m/s
30. Impact speed approximately 43.48 m/s
1.
In Problems 2 through 10 we give the approximation to
2.
3.
4.
5.
6.
7.
8.
9.
10.
11. Solution:
| x | y |
y |
Exact y |
|---|---|---|---|
| 0.0 | 1.00000 | 1.00000 | 1.00000 |
| 0.2 | 0.77860 | 0.77860 | 0.77860 |
| 0.4 | 0.50818 | 0.50818 | 0.50818 |
| 0.6 | 0.17789 | 0.17788 | 0.17788 |
| 0.8 | |||
| 1.0 |
In Problems 12 through 16 we give the final value of x, the corresponding Runge-Kutta approximations with
12.
13.
14.
15.
16.
In Problems 17 through 24 we give the final value of x and the corresponding values of y with
17.
18.
19.
20.
21.
22.
23.
24.
25. With both step sizes
26. With both step sizes
27. With successive step sizes
28. With successive step sizes
29. Time aloft: approximately 9.41 seconds
30. Time aloft: approximately 9.41 seconds
1.
2.
3.
4.
5. Inconsistent—no solution
6. Inconsistent—no solution
7.
8.
9.
10.
11.
12.
13.
14.
15. Inconsistent—no solution
16. Inconsistent—no solution
17. Inconsistent—no solution
18. Inconsistent—no solution
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31. Suggestion: The two lines both pass through the origin.
32. Suggestion: Two distinct planes in space either are parallel or intersect in a straight line.
33. (a) No solution (b) A unique solution (c) No solution (d) No solution (e) A unique solution (f) Infinitely many solutions
34. (a) No solution (b) Infinitely many solutions (c) No solution (d) No solution (e) Infinitely many solutions (f) A unique solution
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15. Inconsistent—no solution
16. Inconsistent—no solution
17.
18.
19.
20.
21.
22.
23. (a) None (b)
24. (a)
25. (a)
26. (a) All k (b) None (c) None
27. (a) None (b)
28. No solution unless
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25. Inconsistent—no solution
26. Inconsistent—no solution
27.
28.
29.
30.
31. The sequence
33.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12. Neither product matrix AB nor BA is defined.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
29.
31. (a)
33.
35.
37.
39.
43.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
1. 60
2. 4
3.
4.
5. 120
6. 60
7. 0
8. 25
9. 30
10. 7
11. 40
12. 10
13. 78
14.
15.
16. 84
17. 8
18. 135
19. 39
20. 79
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
36.
38.
40.
1.
2. 9,
3.
4.
5. Linearly dependent
6. Linearly dependent
7. Linearly dependent
8. Linearly dependent
9.
10.
11.
12.
13.
14.
15. Linearly dependent
16. Linearly dependent
17. Linearly dependent
18. Linearly dependent
19. Linearly dependent;
20. Linearly dependent;
21. Linearly dependent;
22. Linearly dependent
23. Linearly dependent
24. Linearly dependent
25.
26.
27.
28.
1. W is a subspace of
2. W is a subspace of
3. W is not a subspace of
4. W is not a subspace of
5. W is a subspace of
6. W is a subspace of
7. W is not a subspace of
8. W is a subspace of
9. W is not a subspace of
10. W is not a subspace of
11. W is a subspace of
12. W is not a subspace of
13. W is not a subspace of
14. W is not a subspace of
15.
16.
17.
18.
19.
20.
21.
22.
1. Linearly dependent
2. Linearly dependent
3. Linearly dependent
4. Linearly dependent
5. Linearly dependent
6. Linearly dependent
7. Linearly dependent
8. Linearly dependent
9.
10.
11.
12.
13. w cannot be expressed as a linear combination of
14. w cannot be expressed as a linear combination of
15.
16.
17. The vectors
18.
19. The vectors
20. The vectors
21.
22.
1. The vectors
2. The vectors
3. The given vectors do not form a basis for
4. The given vectors do not form a basis for
5. The three vectors
6. The four given vectors form a basis for
7. The three given vectors form a basis for
8. The three given vectors form a basis for
9. The plane
10. The plane
11. The line is a 1-dimensional subspace of
12. Hence the subspace consisting of all such vectors is 3-dimensional with basis consisting of the vectors
13. The subspace consisting of all such vectors is 2-dimensional with basis consisting of the vectors
14. The subspace consisting of all such vectors is 2-dimensional with basis consisting of the vectors
15. The solution space of the given system is 1-dimensional with basis consisting of the vector
16. The solution space of the given system is 1-dimensional with basis consisting of the vector
17. The solution space of the given system is 2-dimensional with basis consisting of the vectors
18. The solution space of the given system is 2-dimensional with basis consisting of the vectors
19. The solution space of the given system is 2-dimensional with basis consisting of the vectors
20. The solution space of the given system is 2-dimensional with basis consisting of the vectors
21. The solution space of the given system is 2-dimensional with basis consisting of the vectors
22. The solution space of the given system is 2-dimensional with basis consisting of the vectors
23. The solution space of the given system is 1-dimensional with basis consisting of the vector
24. The solution space of the given system is 3-dimensional with basis consisting of the vectors
25. The solution space of the given system is 3-dimensional with basis consisting of the vectors
26. The solution space of the given system is 2-dimensional with basis consisting of the vectors
1. Row basis: The first and second row vectors of E. Column basis: The first and second column vectors of A.
2. Row basis: The first and second row vectors of E. Column basis: The first and second column vectors of A.
3. Row basis: The first and second row vectors of E. Column basis: The first and second column vectors of A.
4. Row basis: The three row vectors of E. Column basis: The first three column vectors of A.
5. Row basis: The three row vectors of E. Column basis: The first, second, and fourth column vectors of A.
6. Row basis: The three row vectors of E. Column basis: The first, second, and fourth column vectors of A.
7. Row basis: The first two row vectors of E. Column basis: The first two column vectors of A.
8. Row basis: The first three row vectors of E. Column basis: The first, second, and fourth column vectors of A.
9. Row basis: The first three row vectors of E. Column basis: The first three column vectors of A.
10. Row basis: The first three row vectors of E. Column basis: The first, second, and fourth column vectors of A.
11. Row basis: The first three row vectors of E. Column basis: The first, second, and fifth column vectors of A.
12. Row basis: The first three row vectors of E. Column basis: The first, second, and fifth column vectors of A.
13. Linearly independent:
14. Linearly independent:
15. Linearly independent:
16. Linearly independent:
17. Basis vectors:
18. Basis vectors:
19. Basis vectors:
20. Basis vectors:
21. The first and second equations are irredundant.
22. The first and second equations are irredundant.
23. The first, second, and fourth equations are irredundant.
24. The first, second, and fifth equations are irredundant.
1. Yes, the three vectors are mutually orthogonal.
2. Yes, the three vectors are mutually orthogonal.
3. Yes, the three vectors are mutually orthogonal.
4. Yes, the three vectors are mutually orthogonal.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
1. It is a subspace.
2. It is a subspace.
3. It is a not subspace.
4. It is a not subspace.
5. It is a subspace.
6. It is a not subspace.
7. It is a not subspace.
8. It is a subspace.
9. It is a not subspace.
10. It is a subspace.
11. It is a subspace.
12. It is a not subspace.
13. The functions sin x and cos x are linearly independent.
14. The funtions
15. The functions
16. The three given polynomials are linearly dependent.
17. The three given trigonometric functions are linearly dependent.
18. The two given linear combinations of sin x and cos x are linearly independent.
19.
20.
21.
22.
23. The solution space is 3-dimensional with basis
24. The solution space is 4-dimensional with basis
25. The solution space is 2-dimensional with basis
26. The solution space is 2-dimensional with basis
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
21. Linearly independent
22. Linearly independent
23. Linearly independent
25. Linearly dependent
25. Linearly independent
26. Linearly independent
28.
29. There is no contradiction because if the given differential equation is divided by
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49. The high point is
50.
52.
53.
54.
55.
56.
1.
2.
3.
4.
5.
6.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
38.
39.
40.
41.
42.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
44. (a)
45.
46.
47.
48.
49.
52.
53.
54.
55.
56.
57.
58.
1. Frequency: 2 rad/s (
2. Frequency: 8 rad/sec (
3. Amplitude: 2m; frequency: 5 rad/s; period:
4. (a)
6. About 7.33 mi
7. About 10450 ft
8. 29.59 in.
10. Amplitude: 100 cm; period: about 2.01 sec
11. About 3.8 in.
13. (a)
14. (a)
15.
16.
17.
18.
19.
20.
21.
22. (b) The time-varying amplitude is
23. (a)
34. Damping constant:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43. (b)
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
58.
59.
60.
61.
62.
1.
2.
3.
4.
5.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23. (a) Natural frequency:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
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18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
40. We find that
41. We find that
1.
2.
3.
4.
5.
6.
7.
8.
9. The double eigenvalue
10. The double eigenvalue
11. The double eigenvalue
12. The double eigenvalue
13.
14.
15.
16.
17.
18.
19.
20.
21. The triple eigenvalue
22. The triple eigenvalue
23. The eigenvalues
24. The eigenvalues
25.
26.
27. The eigenvalues
28. The eigenvalues
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38. If
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
1.
3.
5.
7.
9.
11.
13.
15.
17.
19.
21.
23.
25.
27.
29.
31.
32.
33. (a)
1.
2.
3. General solution
4.
5.
6. General solution
7.
8.
9. General solution
10.
11. General solution
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
The maximum amount of salt everin tank 3 is
32.
The maximum amount of salt ever in tank 3 is
33.
The maximum amount of saltever in tank 3 is
34.
The maximum amount of salt ever in tank 3 is
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
Note that phase portraits for Problems 1-16 are found in the answers for Section 7.2.
1. Saddle point (real eigenvalues of opposite sign)
2. Saddle point (real eigenvalues of opposite sign)
3. Saddle point (real eigenvalues of opposite sign)
4. Saddle point (real eigenvalues of opposite sign)
5. Saddle point (real eigenvalues of opposite sign)
6. Improper nodal source (distinct positive real eigenvalues)
7. Saddle point (real eigenvalues of opposite sign)
8. Center (pure imaginary eigenvalues)
9. Center (pure imaginary eigenvalues)
10. Center (pure imaginary eigenvalues)
11. Spiral source (complex conjugate eigenvalues with positive real part)
12. Spiral source (complex conjugate eigenvalues with positive real part)
13. Spiral source (complex conjugate eigenvalues with positive real part)
14. Spiral source (complex conjugate eigenvalues with positive real part)
15. Spiral source (complex conjugate eigenvalues with positive real part)
16. Improper nodal sink (distinct negative real eigenvalues)
17. Center; pure imaginary eigenvalues
18. Improper nodal source; distinct positive real eigenvalues;
19. Saddle point; real eigenvalues of opposite sign;
20. Spiral source; complex conjugate eigenvalues with positive real part
21. Proper nodal source; repeated positive real eigenvalue with linearly independent eigenvectors
22. Parallel lines; one zero and one negative real eigenvalue
23. Spiral sink; complex conjugate eigenvalues with negative real part
24. Improper nodal sink; distinct negative real eigenvalues;
25. Saddle point; real eigenvalues of opposite sign;
26. Center; pure imaginary eigenvalues
27. Improper nodal source; distinct positive real eigenvalues;
28. Spiral sink; complex conjugate eigenvalues with negative real part
1. The natural frequencies are
2. The natural frequencies are
3. The natural frequencies are
4. The natural frequencies are
5. The natural frequencies are
6. The natural frequencies are
7. The natural frequencies are
8.
9.
10.
11. (a) The natural frequencies are
12. The system’s three natural modes of oscillation have (1) natural frequency
13. The system’s three natural modes of oscillation have (1) natural frequency
15.
We have a superposition of two oscillations with the natural frequencies
20.
21.
22.
23.
24. (a)
27.
28.
29.
1. Repeated eigenvalue
2. Repeated eigenvalue
3. Repeated eigenvalue
4. Repeated eigenvalue
5. Repeated eigenvalue
6. Repeated eigenvalue
7. Eigenvalues
8. Eigenvalues
9. Eigenvalues
10. Eigenvalues
11. Triple eigenvalue
12. Triple eigenvalue
13. Triple eigenvalue
14. Triple eigenvalue
15. Triple eigenvalue
16. Triple eigenvalue
17. Triple eigenvalue
18. Triple eigenvalue
19. Double eigenvalues
20. Eigenvalue
21. Eigenvalue
22. Eigenvalue
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
In Problems 37, 39, 41, 43, and 45 we give a nonsingular matrix Q and a Jordan-form matrix J such that
The format for the first eight answers is this:
1.
2.
3.
4.
5.
6.
7.
8.
9. At
10. At
11. At
12. We solved
13. Runge-Kutta,
14. Runge-Kutta,
15. Runge-Kutta,
16. At
17. At
18. Just under
19. Approximately 253 ft/s
20. Maximum height: about 1005 ft, attained in about 5.6 s; range: about 1880 ft; time aloft: about 11.6 s
21. Runge-Kutta with
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
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15.
16.
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20.
21.
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23.
24.
25.
26.
27.
28.
29.
30.
33.
35.
36.
37.
38.
39.
40.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15. (a)
16. (a)
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
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29.
30.
31.
32.
33.
34.
1.
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20.
21.
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23.
24.
25.
26.
27.
28.
29.
30.
31.
32. With
33.
34. With
35.
36. With
37.
38. With
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
1. 6.1.14
2. 6.1.16
3. 6.1.19
4. 6.1.13
5. 6.1.12
6. 6.1.18
7. 6.1.15
8. 6.1.17
9. Equilibrium solutions
10. Equilibrium solution
11. Equilibrium solutions
12. Equilibrium solution
13. Solution
14. Solution
15. Solution
16. Solution
17. Solution
18. Solution
19. Solution
20. Solution
23. The origin and the circles
24. The origin and the hyperbolas
25. The origin and the ellipses
26. The origin and the ovals of the form
1. Asymptotically stable node
2. Unstable improper node
3. Unstable saddle point
4. Unstable saddle point
5. Asymptotically stable node
6. Unstable node
7. Unstable spiral point
8. Asymptotically stable spiral point
9. Stable, but not asymptotically stable, center
10. Stable, but not asymptotically stable, center
11. Asymptotically stable node: (2, 1)
12. Unstable improper node:
13. Unstable saddle point: (2, 2)
14. Unstable saddle point: (3, 4)
15. Asymptotically stable spiral point: (1, 1)
16. Unstable spiral point: (3, 2)
17. Stable center:
18. Stable, but not asymptotically stable, center:
19. (0, 0) is a stable node. Also, there is a saddle point at (0.67, 0.40).
20. (0, 0) is an unstable node. Also, there is a saddle point at
21. (0, 0) is an unstable saddle point. Also, there is a spiral sink at
22. (0, 0) is an unstable saddle point. Also, there are nodal sinks at
23. (0, 0) is a spiral sink. Also, there is a saddle point at
24. (0, 0) is a spiral source. No other critical points are visible.
25. Theorem 2 implies only that (0, 0) is a stable sink—either a node or a spiral point. The phase portrait for
26. Theorem 2 implies only that (0, 0) is an unstable source. The phase portrait for
27. Theorem 2 implies only that (0, 0) is a center or a spiral point, but does not establish its stability. The phase portrait for
28. Theorem 2 implies only that (0, 0) is a center or a spiral point, but does not establish its stability (though in the phase portrait it looks like a likely center). The phase portrait for
29. There is a saddle point at (0, 0). The other critical point (1, 1) is indeterminate, but looks like a center in the phase portrait.
30. There is a saddle point at (1, 1) and a spiral sink at
31. There is a saddle point at (1, 1) and a spiral sink at
32. There is a saddle point at (2, 1) and a spiral sink at
37. Note that the differential equation is homogeneous.
1. Linearization at (0, 0):
3. Linearization at (75, 50):
5. The characteristic equation is
7. The characteristic equation is
Phase plane portrait for the nonlinear system in Problems 4-7:
9. The characteristic equation is
10. The characteristic equation is
Phase plane portrait for the nonlinear system in Problems 8-10:
12. The characteristic equation is
13. The characteristic equation is
15. The characteristic equation is
17. The characteristic equation is
19. The characteristic equation is
21. The characteristic equation is
22. The characteristic equation is
24. The characteristic equation is
25. The characteristic equation is
26. Naturally growing populations in competition Critical points: nodal source (0, 0) and saddle point (3, 2) Nonzero coexisting populations
27. Naturally declining populations in cooperation Critical points: nodal sink (0, 0) and saddle point (3, 2) Nonzero coexisting populations
28. Naturally declining predator, naturally growing prey population Critical points: saddle point (0, 0) and apparent stable center (4, 8)
Nonzero coexisting populations
29. Logistic and naturally growing populations in competition Critical points: nodal source (0, 0), nodal sink (3, 0), and saddle point (2, 2) Nonzero coexisting populations
30. Logistic and naturally declining populations in cooperation Critical points: saddle point (0, 0), nodal sink (3, 0) and saddle point (5, 4) Nonzero coexisting populations
31. Logistic prey, naturally declining predator population Critical points: saddle points (0, 0) and (3, 0), spiral sink (2, 4) Nonzero coexisting populations
32. Logistic populations in cooperation
Critical points: nodal source (0, 0), saddle points (10, 0) and (0, 20), nodal sink (30, 60)
Nonzero coexisting populations
33. Logistic prey and predator populations Critical points: nodal source (0, 0), saddle points (30, 0) and (0, 20), nodal sink (4, 22)
Nonzero coexisting populations
34. Logistic prey and predator populations Critical points: nodal source (0, 0), saddle points (15, 0) and (0, 5), spiral sink (10, 10)
Nonzero coexisting populations
1. Eigenvalues:
2. Eigenvalues: 1, 3; unstable node
3. Eigenvalues:
4. Eigenvalues:
5. Critical points:
6. Critical points:
7. Critical points:
8. Critical points:
9. If n is odd then
10. If n is odd then
11.
12. Unstable saddle points at (2, 0) and
13. Unstable saddle points at (2, 0) and
14. Stable centers at (2, 0) and
15. A stable center at (0, 0) and an unstable saddle point at (4, 0)
16. Stable centers at
17. (0, 0) is a spiral sink.
18. (0, 0) is a spiral sink; the points
19. (0, 0) is a spiral sink.
20.
1.
2.
3.
4.
5.
6.
7.
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23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
37.
38.
39. Figure 10.2.8 shows the graph of the unit staircase function.
1.
2.
3.
4.
5.
6.
7.
8.
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24.
1.
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33.
34.
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36.
37.
38.
1.
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33.
34.
1.
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6.
7.
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15.
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17.
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20.
21.
28.
31.
32.
33.
34.
35.
36.
37.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
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21.
22.
23. As
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
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20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
33. The following figure shows the interlaced zeros of the 4th and 5th Hermite polynomials.
34. The figure below results when we use
1. Ordinary point
2. Ordinary point
3. Irregular singular point
4. Irregular singular point
5. Regular singular point;
6. Regular singular point;
7. Regular singular point;
8. Regular singular point;
9. Regular singular point
10. Regular singular point
11. Regular singular points
12. Irregular singular point
13. Regular singular points
14. Irregular singular points
15. Regular singular point
16. Irregular singular point
17.
18.
19.
20.
21.
22.
23.
24.
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31.
32.
33.
34.
5.
10. 3
11.
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30.